How often do the hands form a straight line?

Question

The hour and minute hands of a clock lie along a single straight line whenever they overlap (angle 0) or point in exactly opposite directions (angle 180). How many such moments are there in 24 hours?

Accepted Answer

44 moments. Overlaps. In 12 hours the minute hand laps the hour hand 11 times, so the hands overlap 11 times in 12 hours and 22 times in 24 hours. Oppositions. During each of those 11 gained laps the minute hand is exactly half a lap ahead once, so the hands are opposite 11 times in 12 hours and 22 times in 24 hours. Total. 22 + 22 = 44. Equivalently, the relative angle between the hands sweeps through a full circle 22 times a day, and each sweep passes 0 degrees once and 180 degrees once. A neat way to see it. Think of the minute hand's angle relative to the hour hand as a single pointer spi…

How often do the hands form a straight line?

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